Lesson plan
Semicomputable function
In computability theory, a semicomputable function is a partial function f : \mathbb{Q} \rightarrow \mathbb{R} that can be approximated either from above or from below by a computable function. More precisely a partial function f : \mathbb{Q} \rightarrow \mathbb{R} is upper semicomputable, meaning it can be approximated from above, if there exists a computable function \phi(x,k) : \mathbb{Q} \times \mathbb{N} \rightarrow \mathbb{Q} , where x is the desired parameter for f(x) and k is the level of approximation, such that: * \lim_{k \rightarrow \infty} \phi(x,k) = f(x) * \forall k \in \mathbb{N} : \phi(x,k+1) \leq \phi(x,k) Completely analogous a partial function f : \mathbb{Q} \rightarrow \mathbb{R} is lower semicomputable if and only if -f(x) is upper semicomputable or equivalently if there exists a computable function \phi(x,k) such that: * \lim_{k \rightarrow \infty} \phi(x,k) = f(x) * \forall k \in \mathbb{N} : \phi(x,k+1) \geq \phi(x,k) If a partial function is both upper and lower semicomputable it is called computable.
Article summary
In computability theory, a semicomputable function is a partial function f : \mathbb{Q} \rightarrow \mathbb{R} that can be approximated either from above or from below by a computable function. More precisely a partial function f : \mathbb{Q} \rightarrow \mathbb{R} is upper semicomputable, meaning it can be approximated from above, if there exists a computable function \phi(x,k) : \mathbb{Q} \times \mathbb{N} \rightarrow \mathbb{Q} , where x is the desired parameter for f(x) and k is the level of approximation, such that: * \lim_{k \rightarrow \infty} \phi(x,k) = f(x) * \forall k \in \mathbb{N} : \phi(x,k+1) \leq \phi(x,k) Completely analogous a partial function f : \mathbb{Q} \rightarrow \mathbb{R} is lower semicomputable if and only if -f(x) is upper semicomputable or equivalently if there exists a computable function \phi(x,k) such that: * \lim_{k \rightarrow \infty} \phi(x,k) = f(x) * \forall k \in \mathbb{N} : \phi(x,k+1) \geq \phi(x,k) If a partial function is both upper and lower semicomputable it is called computable.
Key claims
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Discussion questions
- Summarise the article in your own words. Which single claim would you verify first, and why?
- Which claim in this article is most likely to be revised as the sources update? How would you check its current status?
- If two sources disagreed about one of these facts, what would you need in order to trust one over the other?
Bibliography
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Key claims and bibliography are generated from the same tracked claims and sources as the article page.